Chapter 14 गुणनखंडन Ex 14.1

प्रश्न 1.
दिए हुए पदों में सार्वगुणनखण्ड ज्ञात कीजिए –

हल:
1. 12x = 12 × x
36 = 12 x 3
∴ सार्व गुणनखण्ड = 12

2. 2y = 2 x y
22xy = 2 x 11 × x × y
∴ सार्व गुणनखण्ड = 2 x y = 2y

3. 14pq = 2 x 7 x p x q
28 p²q² = 2 x 2 x 7 x p x p x q x q
∴ सार्व गुणनखण्ड = 2 x 7 x p x q = 14pq

4. 2x = 2 × x × 1
3x² = 3 × x × x × 1
4 = 2 x 2 x 1
∴ सार्व गुणनखण्ड = 1

5. 6abc = 2 x 3 x 4 x 6 x c
24ab² =2 x 2 x 2 x 3 x a x b x b
12a²b = 2 x 2 x 3 x a x a x b
∴ सार्व गुणनखण्ड = 2 x 3 x a x b = 6ab

6. 16x³ = 2 × 2 × 2 × 2 × x × x × x
4x² = (-1) × 2 × 2 × x × x
32x = 2 × 2 × 2 × 2 × 2 × x
∴ सार्व गुणनखण्ड = 2 × 2 × x = 4x

7. 10pq = 2 x 5 x p x q
20qr = 2 x 2 x 5 x q x r
30pr = 2 x 3 x 5 x p x r
∴ सार्व गुणनखण्ड = 2 x 5 = 10

8. 3x²y³ = 3 × x × x × y × y × y
10x³y² = 2 × 5 × x × x × x × y × y
6x²y²z = 2 × 3 × x × x × y × y × z
∴ सार्व गुणनखण्ड = x × x × y × y = xy²

प्रश्न 2.
निम्नलिखित व्यंजकों के गुणनखण्ड कीजिए –

हल:
1. 7x – 42 = 7 × x – 2 × 3 × 7
= 7(x – 2 x 3)
= 7 (x – 6)

2. 6p – 12q = 2 x 3 x p – 2 x 2 x 3 x q
= 2 x 3 (p – 2 x q)
= 6 (p – 24)

3. 7a² + 14a = 7 x a x a + 2 x 7 x a
= 7 x a x (a + 2)
= 7a (a + 2)

4. – 16z + 20z³
= -2 x 2 x 2 x 2 x z + 2 x 2 x 5 x z x z x z
= 2 x 2 x z x (- 2 x 2 + 5 x z x z)
= 4z (- 4 + 57²)

5. 20l²m + 30alm
= 2 x 2 x 5 x 1 x 1 x m + 2 x 3 x 5 x a x l x m
= 2 x 5 x 1 x m (2 x 1 + 3 x a)
= 10lm (21+ 3a)

6. 5x²y – 15xy²
= 5 × x × x × y – 3 × 5 × x × y × y
= 5 × x × y × (x – 3 × y)
= 5xy (x – 3y)

7. 10a² – 15b² + 20c²
= 2 x 5 x a x a – 3 x 5 x b x b + 2 x 2 x 5 x c x c
= 5 (2 x a x a – 3 x b x b + 2 x 2 x c x c)²
=5 (2a² – 3b² – 4c)

8. -4a² + 4ab – 4ac
= – 2 x 2 x a x a + 2 x 2 x a x b – 2 x 2 x a x c
= 2 x 2 x a x (- a + b – c)
= 4a (- a + b – c)

9. xyz + xyz + xyz²
= x × x × y × z + x × y × y × z + x × y × z × z
= x × y × z (x + y + z)
=xyz (x + y + z)

10. ax²y + bxy² + cxyz
= a × x × x × y + b × x × y × y + c × x × y × z
= x × y × (a × x + b × y + c × z)
= xy (ax + by + cz)

प्रश्न 3.
गुणनखण्ड कीजिए –

हल:
1. x² + xy + 8x + 8y = (x² + xy) + (8x + 8y)
= x (x + y) + 8 (x + y)
= (x + y) (x + 8)

2. 15xy – 6x + 5y – 2 = (15xy – 6x) + (5y – 2)
= 3x (5y – 2) + 1 (5y – 2)
= (5y – 2) (3x + 1)

3. ax + bx – ay – by = (ax + bx) – (ay + by)
= x (a + b) – y (a + b)
= (a + b) (x – y)

4. 15pq + 15 + 9q + 25p = (15pq + 9q) + (25p + 15)
(पुनः समूहन करने पर)
= 3q (5p + 3) + 5 (5p + 3)
= (5p + 3) (3q + 5)

5. z – 7 + 7xy – xyz = z – 7 – xyz + 7xy
(पुनः समूहन करने पर)
= 1 (z – 7) – xy (z – 7)
= (z – 7) (1 – xy)